Limit State Moment Functions for Laterally Loaded Single Piles in ...

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Limit State Moment Functions for Laterally Loaded Single Piles in Local OC Clay

Aşırı Konsolide Kil Zeminlerdeki Yanal Yüklü Kazıklar için Limit Durum Moment Fonksiyonları

Gökhan İMANÇLI*, Mehmet Rifat KAHYAOĞLU ve Gürkan ÖZDEN

Dokuz Eylül Üniversitesi, Department of Civil Engineering, Faculty of Engineering, 35160, İzmir

Geliş Tarihi/Received : 06.01.2009, Kabul Tarihi/Accepted : 16.06.2009

ABSTRACTPerformance level of piles is usually expressed with the maximum head deflection and bending mo-ment. Uncertainties in the performance of a pile originates from many factors such as spatial varia-tion of soil properties, inadequate soil investigations, errors taking place in the determination of soil parameters by various testing methods, and limited calculation models. These uncertainties make it difficult for practicing engineers to effectively select the appropriate method when designing la-terally loaded piles. In this study, the conventional, numerical, two and three dimensional finite ele-ment solutions of free head loading conditions are analyzed for determining the maximum deflecti-on and moment of a laterally loaded single concrete pile in a homogenous fully saturated overcon-solidated local cohesive soil. The local clay has PI and OCR values of 35 and 6.1, respectively. While developing limit state functions via response surface method, maximum bending moment data un-der service load condition computed using 3D finite element analysis, which better represents the three dimensional nature of the soil-structure interaction, was used by considering the pile rigidity.

Keywords : Laterally loaded pile, Oc clay, Pile rigidity, 3-D Finite element analysis, Response surface

method, Limit state moment function.

ÖZETKazıkların performans düzeyleri genellikle maksimum kazık başı deplasmanı ve maksimum eğilme momenti ile ifade edilir. Kazık performansındaki belirsizlikler; zemin özelliklerindeki bölgesel değişiklikler, yetersiz zemin etüd çalışmaları, zemin parametrelerinin belirlenmesinde farklı deney yöntemlerinden oluşabilecek hatalar ve sinirli hesap modelleri gibi etkenlerden kaynaklanabilmek-tedir. Bu belirsizlikler, uygulamacı mühendislerin yanal yüklü kazık dizaynı için kullanacakları uygun yöntemi seçebilmesini zorlaştırmaktadır. Bu çalışmada, suya doygun homojen aşırı konsolide yerel bir kil zemindeki serbest başlı kazıkların yanal yükler altında maksimum kazık başı deplasmanı ve maksimum eğilme momenti değerlerini belirlemek için yapılmış olan konvansiyonel, sayısal ve iki ve üç boyutlu sonlu elemanlar analizleri gerçekleştirilmiştir. Yerel kil zeminin plastisite indisi ve aşırı kon-solidasyon oranı değerleri sırası ile 35 ve 6.1 dir. Tepki yüzeyi yöntemi ile limit durum fonksiyonları geliştirilirken, servis yükleri altında kazığın üç boyutlu davranışını daha iyi yansıtan üç boyutlu sonlu elemanlar analizi ile kazık rijitliği dikkate alınarak belirlenen maksimum moment değerleri kullanılmıştır.

Anahtar Kelimeler : Yanal yüklü kazık, Aşırı konsolide kil, Kazık rijitliği, Üç boyutlu sonlu elemanlar

analizi, Tepki yüzeyi yöntemi, Limit durum moment fonksiyonu.

* Yazışılan yazar/Corresponding author. E-posta adresi/E-mail address : [email protected] (G. İmançlı)

Pamukkale Üniversitesi Mühendislik Bilimleri DergisiCilt 15, Sayı 3, 2009, Sayfa 405-416

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Pamukkale University Journal of Engineering Sciences, Vol. 15, No. 3, 2009

G. İmançlı, M. R. Kahyaoğlu and G. Özden

1. INTRODUCTIONLateral loads have at least the same importance as axial compressive loads on piles and there-fore they must be carefully taken into consid-eration during design. Lateral loading of a pile may be due to ‘active’ loading where external loads are applied at the pile head or due to ‘pas-sive’ loading where lateral movement of the soil induces bending stresses in the pile (Fleming et al., 1994).

Laterally loaded piles must satisfy geotechnical and structural design criteria. In general, geo-technical design criteria dictate pile dimensions (i.e. diameter and length) and pile type. The maximum moment, occurs typically at a depth of 0.1-0.4 times the length of the pile below the surface, in a free-head pile with a horizontal load at the top depends on the relative pile-soil stiffness factor and loading conditions (Hsiung, 2003).

The maximum deflection, on the other hand, usually takes place at the top of the pile. In current design practice where performance based design has become a crucial task in earth-quake prone areas, geotechnical engineers are expected to accurately predict both maximum deflection and moment in the design stage.

Modern analysis methods that allow the lateral load-deflection behavior of piles have become available since late 1960s. Broms method is based primarily on the use of limiting values of soil resistance and solution of the static equilib-rium of the pile. Lateral deflections at working loads have been calculated using the concept of subgrade reaction. For instance, the response of a single pile subjected to lateral loads has been successfully modeled by means of finite difference method (FDM) using nonlinear p–y curves. The p–y analysis requires use of numeri-cal solution algorithms (Wang and Reese, 1993; Reese and Wang, 1996). In the recent decades, the finite element method (FEM), which is more versatile than the finite difference method, has been widely used as the most efficient means for evaluating soil-pile interaction (Randolph, 1981).

Among these approaches, FEM computed load-deflection curves often satisfy the results of laterally loaded pile field tests since they con-sider three dimensional and nonlinear behav-

ior of soil around pile (Tand and Vipulanandan 2008). Although this is the case, 3D modeling of the pile and surrounding soil requires labor intensive work during modeling stage result-ing in high analyses costs which are not usually justified for majority of the projects. There is still need in practical design tools that enable engi-neers to achieve performance based pile analy-sis.

In this study, behavior of both rigid and flex-ible single concrete piles were taken into con-sideration using Broms (1964), Murthy and Subba Rao (1995), p-y curve (Reese, 1985), two and three dimensional finite element methods (Randolph, 1981; Brinkgreve and Broere, 2006). Firstly, the service load level corresponding to allowed lateral pile head deformation is com-puted using Broms method. Rigid and flexible pile cases were then examined by keeping the pile length constant while varying the pile di-ameter and the undrained shear strength of fully saturated OC clay. It was decided that three dimensional FEM better represents complex nature of the problem. Simple yet still satisfac-tory limit state moment functions for laterally loaded piles in homogenous cohesive soils are obtained with response surface method (RSM) using the analysis results of 3D FEM.

2. REVIEW OF PREVIOUS STUDIES

The magnitude of the soil reaction to a laterally deforming pile is a function of the pile deflec-tion, which depends on the pile rigidity and loading conditions. Thus solving the behavior of a pile under lateral loading involves solution of a complex soil-structure-interaction problem.

Lateral loads and moments on a vertical pile are resisted by the flexural stiffness of the pile and mobilization of resistance in the surround-ing soil as the pile deflects. Figure 1 shows the mechanism where the ultimate soil resistance is mobilized to resist a combination of lateral force (P) and moment (M) applied at the top of a free-head pile.

The ultimate lateral resistance Pu and the cor-responding moment Mu can then be related to the ultimate soil resistance pxu. The soil re-sistance against the lateral movement of the pile can be considered in two components; the

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Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, Cilt 15, Sayı 3, 2009


normal force and the shear forces (Briaud and Smith, 1983) as shown in Figure 2.

Lateral capacity of flexible (long) piles is pri-marily dependent on the yield moment of the pile whereas the lateral capacity of rigid (short) piles is mostly dependent on the soil resistance.

Many researchers (e.g. Brinch Hansen, 1961; Broms, 1964; Poulos, 1971; Randolph, 1981; Re-ese, 1985; Murthy and Subba Rao, 1995) have investigated the laterally loaded pile behavior and ultimate lateral resistance to piles in cohe-sive soils.

Figure 1. Mobilization of lateral resistance for a free head laterally loaded rigid pile (Prakash and Sharma, 1990).

Figure 1. Mobilization of lateral resistance for a free head laterally loaded rigid pile (Prakash and Sharma, 1990).

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Methods for calculating lateral resistance of ver-tical piles can be broadly divided into two cat-egories: (a) Methods for calculating ultimate lat-eral resistance, and (b) Methods for calculating acceptable deflection at working lateral load. The latter approaches are usually preferred over the ultimate lateral resistance based methods since soil-structure interaction analyses of pile supported structures require evaluation of de-formation levels.

One of the well-known methods capable of yielding pile sectional forces and pile deforma-tion is due to Broms (1964). In this approach the soil reaction is related to pile deformation by means of horizontal subgrade modulus (kh). Although it offers ready-to-use design charts for free and fixed head piles, the Broms method is only applicable to fully homogenous cohesive and cohesionless soils.

In a theoretically more sound approach, Poulos (1971) proposed a linear analysis methodology based on the theory of elasticity where Mindlin’s governing equations (1936) are integrated. The soil in this case is assumed as an elastic, homo-geneous, isotropic mass having constant elastic parameters E and ν with depth. Also the pile is considered to be a thin vertical strip having with (B), length (L), and constant flexibility (EpIp). The most significant simplification in Poulos ap-proach is that soil and pile are assumed as fully compatible and horizontal shear stresses devel-oped between the soil and the sides of the pile are not taken into account. The lateral behavior of a pile is influenced by the length-to-diameter ratio, L/B, stiffness of the pile and relative stiff-ness of the pile/soil material.

Murthy and Subba Rao (1995) developed ex-plicit relationship between coefficient of sub-grade modulus, nh, and other soil-pile proper-ties based on dimensional analysis principles. Coefficient of subgrade modulus was expressed as a function of various parameters such as stiff-ness and diameter of the pile, unit weight and undrained shear strength of the soil.

Other methods that are based on beam on elas-tic foundation approach are usually considered as derivations of the original Winkler’s method (1867). The method is especially suitable to nu-merical analyses through the discretization of the governing equation (Eq. 1).

where, Ep is modulus of elasticity of the pile; Ip is the moment of inertia of pile section; x is any point along pile depth; kh is the modulus of horizontal subgrade reaction; and y is the pile deformation.

Winkler’s approach has been later adapted to nonlinear soil-pile interaction analyses by defin-ing nonlinear soil-pile deformation curves (p-y) and interaction models for concrete material (Reese, 1985).

Majority of the available laterally loaded pile analysis methods suffer from the disadvantage of not being able to account for the slippage between the soil and laterally deforming pile. Some p-y models have been proposed in the past for considering gap formation and shear stress generation on soil-pile interface (Nogami et al., 1992). However, their utilization requires establishment of complex p-y curves and nu-merical solution algorithms.

During the recent decades, FEM has been wide-ly used by researchers. Randolph (1981) per-formed a parametric study using the 2D FEM for flexible piles in homogeneous soil that was treated as an elastic continuum characterized by a shear modulus (G), Poisson’s ratio (υ) and in soil with stiffness proportional to depth rather than by a coefficient of subgrade reaction.

One of the primary advantages of the FEM is that it can be easily extended to a stratified soil medium by taking material nonlinearity and slippage along the soil-pile interface into ac-count within three dimensions. The FEM, how-ever, is computationally expensive compared to other methods since a large number of finite el-ements with several degrees of freedoms need to be employed in order to accurately model the behavior of the pile.

3. THE SOIL-PILE MODEL

3D FEM using commercially available software, PLAXIS 3D FOUNDATION, was preferred in this study in order to account for soil structure inter-action aspects such as soil nonlinearity, soil-pile slippage and relative soil-pile stiffness. The pile-soil model of this study is given in Figure 3.


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In this model, a free head concrete pile (C30) with an embedment length of 10 m in homoge-neous clay is loaded at the top. The soil profile is assumed to be homogeneous with constant deformation modulus (Eu) and undrained shear strength (cu) values to a depth of 15 m. These two parameters are linked to each other by means of a well known empirical relationship (Eq.2) (Duncan and Buchignani, 1976):

where, Eu is the undrained deformation modu-lus (kPa), Kcu is a factor relating Eu to cu (Figure 4).

The plasticity index and overconsolidation pa-rameters are respectively selected as 35 and 6.1 in order to match properties of Gaziemir clay (Kayalar et al., 1991). In this study the empirical adjustment coefficient, Kcu, is selected as 200 resulting in the variation of elastic modulus be-tween 1x104 and 2x104 kN/m2.

The length-to-diameter ratio of the pile varied between 8.3 and 25 for a constant pile length of 10 m covering a wide variation range for the pile flexibility parameter, β, as defined by Eq. 3 (Hannigan et al., 1998). The pile is considered as long (i.e. flexible) when βL > 2.25, and short (i.e. rigid) for βL < 2.25.

In the above equation, EpIp is the bending stiff-ness of the pile section, kh is the modulus of horizontal subgrade reaction, B and L are the diameter and length of the pile, respectively.

The service load level is defined by factoring the load determined according to Broms Meth-od (Broms, 1964) using the pile diameter, pile length and βL corresponding to the maximum allowable lateral head deflection (y0).

y0, the maximum expected value of which is 3.0 cm in this case, is the allowable deflection cor-responding to the strain (εc) at a stress equal to the maximum stress resulting from the labora-tory stress/strain curve. εc can be also obtained from the established relationship between cu and the deformation modulus Eu. The values of εc have been established with the range of Eu/cu of 50 to 200 for most clays and an average value of εc is recommended as 0.001 (Matlock, 1970).

Broms Method is one of the well-known and easily applicable for practical design method with offered deflection charts for both rigid and flexible piles. In this approach the soil reaction is related to pile deformation by means of hori-zontal subgrade modulus (kh).Typical values of kh within the critical depth for clays have been accumulated by a formula given below (Broms 1964).

Figure 3. Soil-pile model.

Figure 4. Chart for estimating undrained modulus of clay (compiled from Duncan and Buchignani, 1976).

where, qu=the unconfined compressive strength (kPa); B=the diameter of pile (m); n1=0.36 (ac-cording to unconfined compression strength); n2=1.15 (for concrete pile). For overconsolidated clay a value of kh in the range of 2760-16760 kN/m3 is adopted.

For piles in saturated cohesive soils, using di-mensionless length (βL) and maximum allow-able lateral pile head deflection (y0) (i.e. 30 mm), working load Qa (kN) is obtained with the cor-responding value of from Figure 5 without eccentricity of loading (e = 0).

The factoring coefficient is taken as 2.5 as sug-gested in the literature (Hannigan et al., 1998). The service loads are given inFigure 6.

Such a situation can only be analyzed effec-tively by means of three-dimensional finite

element calculations in which proper models are incorporated to simulate soil behavior and soil-structure interaction. A comprehensive computation of the response of the laterally loaded single pile requires 3D modeling of the problem. The constitutive soil model in FEM analyses was the Mohr-Coulomb model. Soil el-ements are assumed to be homogeneous and isotropic. Undrained total stress conditions are taken into consideration. The soil-pile interface strength parameter is set to two-thirds of the corresponding soil strength parameter via the interface parameter (Rinter) so that strength re-duction due to slippage of the soil around the pile is taken into consideration. The pile was modeled as a linear elastic massive circular pile. Details of the soil and pile models are given in Table 1 and Table 2, respectively.

The three dimensional nature of the pile and the surrounding soil was modeled using a fine mesh consisting of 4380 fifteen node wedge elements. The number of finite elements and model dimensions were decided upon numer-ous FEM analyses trials until discretization and boundary effects became negligible. The square working plane, having dimensions of 40 pile di-ameters wide by 40 pile diameters long and 2 pile lengths depth, is selected to eliminate the boundary effects and the pile is positioned at the centre of this plane.

Several FEM runs were performed for differ-ent pairs of pile diameter and undrained shear strength. Pile diameter varied between 40 and 120 cm whereas the variation range for und-rained shear strength was 50 to 200 kPa.

Pile head deformation and bending moment data calculated via the FEM method were found to be in good agreement with those of a previous study (Randolph, 1981). The pile is positioned at the centre of this area. A typical model with the assigned boundary conditions (i.e. constrained in the lateral axis and free to de-form along the vertical axis) and the FEM mesh is shown in Figure 7.

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Figure 5. Lateral deflection of pile at ground surface in cohesive soil (Broms 1964).

Figure 6. Service loads for FEM analyses.


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4. ANALYSIS RESULTS AND DISCUSSIONS

An extensive parametric study were carried out by conventional (Broms, Murthy and Subba Rao), numerical (p-y curve method) and two and three dimensional finite element methods (Randolph, Brinkgreve and Broere) to investi-gate the effects of variation of soil parameters and pile diameters on pile head displacement and maximum moment of a laterally loaded single pile. A free head concrete pile having a 10 meter embedded length in saturated overcon-solidated clay was forced on head by a lateral load.

In the analyses, the pile diameter is varied as 40cm, 60cm, 80cm, 100cm and 120cm. where-

as the undrained shear strength value ranging from 50 to 200 kPa were used. Analysis results of the pile head displacements and maximum moments are presented and discussed as in the following.

Behavior of laterally loaded pile is a function of the pile soil relative stiffness. Piles are either defined as rigid or flexible piles depending on their deformed shape under lateral load. The pile rigidity factor as defined previously can be used while deciding whether a pile is rigid or flexible. It can be noticed in Figure 8 that the transition from rigid to flexible pile behavior oc-curs for the data pair of cu=100kPa and B=60cm.

The variation of maximum moment values with undrained shear strength and pile diameter de-termined by the mentioned methods is given in Figure 9. As expected, as the pile diameter increases the maximum moment on the pile subjected to different service loads increases as well for both flexible and rigid piles. In the case of flexible pile (B=0.4m and B=0.6m where cu>100kPa), the maximum moments deter-mined by the mentioned five different meth-ods, are also the same in spite of increasing undrained shear strength. Nevertheless, in the case of rigid piles (B=0.6m where cu<100 kPa, B=0.8m, 1.0m, 1.2m), the maximum moment values increase parallel to the increasing of un-drained shear strength.

Table 1. Material properties of the homogeneous cohesive soil.

Table 2. Material properties of the piles.

Figure 7. Finite element mesh of the model.

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Figure 8. Pile diameter-pile rigidity relationship at various undrained shear strength value.

Figure 9. Variation of the maximum bending moment with undrained shear strength and pile diameter.

The Figure 9 clearly indicates that among the piles having the same pile diameter, the mo-ments obtained by Broms method have the greatest value as compared to the other meth-ods.

The results of the analysis for pile head dis-placement values for regarding to different un-drained shear strengths and pile diameters are given in Figure 10.


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The head displacement of flexible piles decreas-es while the shear strength increases, whereas that of rigid piles increases. It can be easily seen in the figure that among the piles that have the same pile diameter and in the same soil condi-

tion, the values of pile head displacement ob-tained by 3-D FEM were really greater than the values obtained by the Murthy, p-y and Ran-dolph methods.

5. DEVELOPMENT OF LIMIT STATE MOMENT FUNCTIONS

Contemporary methods for risk assessment and reliability analysis of various geotechnical problems often require development of limit state and performance functions since compu-tational costs can be reduced in this manner by avoiding repeated solutions of governing equa-tions. The multiple linear regressions (MLR) and the response surface methods (RSM) are gener-ally utilized in order to achieve such functions. These approaches have been applied to the reli-ability analysis of laterally loaded piles, retaining systems and differential settlement problems in the past (Tandjiria et al., 2000; Mahedevan and Shi, 2001; Pula, 2007; Park et al., 2007).

RSM is a collection of statistical and mathemati-cal techniques useful for developing, improving and optimizing process including reduction of variability. In the practical application of RSM, it

is necessary to develop an approximating mod-el for the true response surface. Multiple regres-sion is a collection of statistical techniques use-ful for building the types of empirical models required in RSM. The key ideas in the classical RSM were developed using linear polynomial models, mainly first degree and second degree models, with continuous response variables as-sumed, for the most part, to be independently and normally distributed with constant error variances. A first order response surface model that might describe the relationship between Y and X1, X2,…, Xp is formulated as a linear model

where β0, β1, β2,....., βp are constants referred to as the model regression coefficients, p is the re-gressor variable, and ε is a random disturbance or error.

In general, response variable Y may be related to p regressor variables which model is called a

Figure 10. Pile head displacement parameter as a function of undrained shear strength and pile diameter.

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multiple linear regression model (MLR). MLR at-tempts to model the relationship between two or more independent variables and a response or dependent variable by fitting a linear equa-tion to observed data. Every value of the inde-pendent variable X is associated with a value of the dependent variable Y. (Myers and Mont-gomery, 2002)

Bending moment is the key parameter for struc-tural design of laterally loaded piles. Deforma-tion criterion for pile performance, however, is implicitly taken into consideration by defining the service lateral load as the forty percent of the load resulting in 30 mm of pile head dis-placement which is generally accepted.

In this study, maximum bending moment data computed using FEM, which better represented three dimensional nature of the problem upon evaluation of laterally loaded pile analyses re-sults, was used while developing limit state functions via RSM where the pile rigidity is tak-en into consideration. RSM analyses were exam-ined to find which of the soil and pile proper-ties correlate well with the maximum moment values for both rigid and flexible piles. The un-drained shear strength (cu), which is linearly correlated with elastic modulus of soil (Eu), and the pile diameter (B) were used as independent variables.

The following Limit State Moment Functions have been developed for both rigid (short) and flexible (long) piles (Eq.6 and Eq.7):

where the pile diameter (B) is in units of meter, and undrained shear strength (cu) is in units of kN/m2.

The limit state functions as proposed here yield the maximum moment values for both rigid and flexible piles satisfactorily. In these linear relations, the coefficients of correlation (R) for rigid and flexible pile cases were determined as 0.97 and 0.99, respectively. The variation of the

maximum moments estimated using limit state functions are given in Figure 11.

The computed maximum moment via pro-posed limit state moment functions correlated well with the results of full scale lateral load tests on rigid piles bearing in stiff to very stiff clay at the National Geotechnical Experimenta-tion Site located at the University of Houston in January 1997 (Tand and Vipulanandan 2008). However, additional research needs to be per-formed to better determine the maximum mo-ment on piles.

Statistics, reliability analysis and risk estimates are important tools to account for uncertainties and to make properly decisions. Hence, proba-bilistic analysis should be used when uncer-tainty in parameters may govern the results of the analysis. The probabilistic analyses quantify likelihood of failure from the statistical varia-tions of input parameters or model errors. For instance, the performance functions regarding limit state functions and ultimate design crite-rion can be utilized in limit state-based reliabil-ity analysis which are commonly used the first-order and second-order reliability methods to take into account measurement uncertainties of soil parameters and pile diameter.

a) Rigid piles.


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The analytical models which are most common-ly used in practice to predict the maximum mo-ment, as well as acceptable deflection of laterally loaded piles should be calibrated with full-scale load tests to ascertain by pile designers because of the geotechnical uncertainties. Nevertheless, full-scale load tests are generally not performed by designers due to the difficulties and expen-siveness except research projects. On the other hand model uncertainties can be significant due to the lack of knowledge and inability to model precisely, and soil-structure interaction behavior. The limit state functions proposed in this study can be used for consideration of un-certainties in undrained shear strength and pile diameter parameters.

6. CONCLUSIONS

Numerous analyses were carried out by con-ventional, numerical and two and three dimen-sional finite element methods to investigate the effects of pile diameter and undrained shear strength on the response of laterally loaded single pile. This study in this paper leads the fol-lowing conclusions.

• The maximum bending moment on the flexible pile is approximately constant while increasing undrained shear strength; whereas on the rigid pile it increases par-allel to the increasing of undrained shear strength. In another words, bending mo-ment is not as sensitive to undrained shear strength for flexible piles.

• The head displacement of flexible piles de-creases whereas that of rigid piles increas-es; while the undrained shear strength is increasing.

• Analysis results shows that limit state mo-ment functions can be effectively evalu-ated taking soil-structure-interaction into account. Maximum bending moment is expressed as a linear function of und-rained shear strength and pile diameter. The proposed limit state functions should be considered as linear solution planes of the lateral pile response bounded by the prescribed pile head displacement, varia-tion range of the undrained shear strength and pile diameter.

• The relative soil-pile stiffness factor should be determined while utilizing the moment limit state function.

b) Flexible piles

Figure 11. Variation of the estimated maximum pile moment with undrained shear strength and pile

diameter.

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